Segment diffusion and nuclear magnetic resonance spin-lattice relaxation of polymer chains confined in tubes: Analytical treatment and Monte Carlo simulation of the crossover from Rouse to reptation dynamics (original) (raw)
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Polymer chain dynamics and NMR
Advances in Polymer Science
The universal features of polymer dynamics are specifically represented by laws for (anomalous) segment diffusion and chain relaxation modes. Nuclear magnetic resonance (NMR)-based techniques provide direct access to these phenomena. This in particular refers to NMR relaxation and diffusion studies. Methods suitable for this purpose are described in detail. Three basic classes of polymer dynamics models, namely the Rouse model, the tube/reptation model, and the renormalized Rouse models are outlined and discussed with respect to predictions for NMR measurands. A wealth of experimental NMR data are reviewed and compared with predictions of the model theories. It is shown that characteristic features of all three types of models can be verified in great detail provided that the model premisses are suitably mimicked in the experiments. Rouse dynamics is shown to be relevant for polymer melts with molecular weights below the critical value and for solutions of diminished entanglement effect. Features specific for the renormalized Rouse model reveal themselves in the form of high-and low-mode-number limits of the spin-lattice relaxation dispersion. These results are considered to mirror the analytical structure of the Generalized Langevin Equation. Finally, anomalous-diffusion and relaxation laws characteristic for the tube/reptation model can be perfectly reproduced in experiment if the polymer chains are confined in a nanoporous, solid matrix whereas bulk melts are not in accord with these predictions. The dynamics of chains confined in artificial tubes can be treated analytically assuming a harmonic radial potential for the polymer/wall interaction. These results derived for a real tube closely render the characteristic features of the original Doi/Edwards model predicted for a fictitious tube.
Solid State Nuclear Magnetic Resonance, 2009
The Mori-Zwanzig projection operator technique was employed to derive the effective Hamiltonian for spin-segment coupling. The fluctuations of this operator are responsible for spin-lattice relaxation in polymer chains. In detail, dipolar interaction of spins is rigorously analyzed by components representing fluctuations of the Kuhn segment end-to-end vectors and local fluctuations on a length scale shorter than the root mean square Kuhn segment length. The former correspond to the usual coarse-grain picture of polymer chain mode theories. It is shown that these non-local chain modes dominate proton spin-lattice relaxation dispersion of flexible polymers at frequencies up to about 10 8 Hz. A corresponding evaluation of experimental data for polybutadiene melts is presented.
Polymer chain dynamics under nanoscopic confinements
Magnetic Resonance Imaging, 2005
It is shown that the confinement of polymer melts in nanopores leads to chain dynamics dramatically different from bulk behavior. This so-called corset effect occurs both above and below the critical molecular mass and induces the dynamic features predicted for reptation. A spinodal demixing technique was employed for the preparation of linear poly(ethylene oxide) (PEO) confined to nanoscopic strands that are in turn embedded in a quasi-solid and impenetrable methacrylate matrix. Both the molecular weight of the PEO and the mean diameter of the strands were varied to a certain degree. The chain dynamics of the PEO in the molten state was examined with the aid of field-gradient NMR diffusometry (time scale, 10 À2-10 0 s) and field-cycling NMR relaxometry (time scale, 10 À9-10 À4 s). The dominating mechanism for translational displacements probed in the nanoscopic strands by either technique is shown to be reptation. On the time scale of spin-lattice relaxation time measurements, the frequency dependence signature of reptation (i.e., T 1~m 3/4) showed up in all samples. A btubeQ diameter of only 0.6 nm was concluded to be effective on this time scale even when the strand diameter was larger than the radius of gyration of the PEO random coils. This corset effect is traced back to the lack of the local fluctuation capacity of the free volume in nanoscopic confinements. The confinement dimension is estimated at which the crossover from confined to bulk chain dynamics is expected.
Progress in Nuclear Magnetic Resonance Spectroscopy, 2017
Field-cycling NMR relaxometry is a well-established technique for probing molecular dynamics in a frequency range from typically a few kHz up to several tens of MHz. For the interpretation of relaxometry data, it is quite often assumed that the spin-lattice relaxation process is of an intra-molecular nature so that rotational fluctuations dominate. However, dipolar interactions as the main type of couplings between protons and other dipolar species without quadrupole moments can imply appreciable inter-molecular contributions. These fluctuate due to translational displacements and to a lesser degree also by rotational reorientations in the short-range limit. The analysis of the inter-molecular proton spin-lattice relaxation rate thus permits one to evaluate self-diffusion variables such as the diffusion coefficient or the mean square displacement on a time scale from nanoseconds to several hundreds of microseconds. Numerous applications to solvents, plastic liquids and polymers will be reviewed. The technique is of particular interest for polymer dynamics since intermolecular spin-lattice relaxation diffusometry bridges the time scales of quasi-elastic neutron scattering and field-gradient NMR diffusometry. This is just the range where model-specific intra-coil mechanisms are assumed to occur. They are expected to reveal themselves by characteristic power laws for the time-dependence of the mean-square segment displacement. These can be favorably tested on this basis. Results reported in the literature will be compared with theoretical predictions. On the other hand, there is a second way for translational diffusion phenomena to affect the spin-lattice relaxation dispersion. If rotational diffusion of molecules is restricted, translational diffusion properties can be deduced even from molecular reorientation dynamics detected by intra-molecular spin-lattice relaxation. This sort of scenario will be relevant for adsorbates on surfaces or polymer segments under entanglement and chain connectivity constraints. Under such conditions, reorientations will be correlated with translational displacements leading to the so-called RMTD relaxation process (reorientation mediated by translational displacements). Applications to porous glasses, protein solutions, lipid bilayers, and clays will be discussed. Finally, we will address the intriguing fact that the various time limits of the segment mean-square displacement of polymers in some cases perfectly reproduce predictions of the tube/reptation model whereas the reorientation dynamics suggests strongly deviating power laws.
The coil–globule transition for a polymer chain confined in a tube: A Monte Carlo simulation
Journal of Chemical Physics, 2000
The behavior of a grafted polymer chain confined in a tube is investigated within a scaling theory substantiated with biased Monte Carlo simulations of a self-avoiding walk ͑SAW͒ on a cubic lattice. All the statistical and thermodynamic properties of the chain follow from the knowledge of the joint distribution P(z,m) giving the probability to observe a length z and a number of contacts m, in a model where the energy of the chain in a given configuration is proportional to m. The analysis is based on the factorization of P(z,m) into the a priori distribution P(z) and the conditional probability P(m͉z) of finding m contacts given that the chain length is z. P(m͉z) is well-approximated by a Gaussian distribution. Taking the variance ͗m 2 ͘Ϫm 2 of this distribution into account, we obtain a nonmean-field expression for the free energy of the confined chain. We show that the coil-globule transition of the confined chain is independent of its size but depends on the pore diameter. Contrary to free, unconfined chains, it is always a continuous transition.
Journal of Chemical Physics, 2010
Proton NMR phenomena such as spin-lattice relaxation, free-induction decays, and solid echoes are analyzed with respect to contributions by intermolecular dipole-dipole interactions in polymer melts. The intermolecular dipole-dipole correlation function is calculated by taking into account the correlation hole effect characteristic for polymer melts. It is shown that the ratio between the intraand intermolecular contributions to NMR measurands depends on the degree of isotropy of chain dynamics anticipated in different models. This, in particular, refers to the tube/reptation model that is intrinsically anisotropic in clear contrast to n-renormalized Rouse models, where no such restriction is implied. Due to anisotropy, the tube/reptation model predicts that the intramolecular contribution to the dipole-dipole correlation function increases with time relative to the intermolecular contribution. Therefore, the intramolecular contribution is expected to dominate NMR measurands by tendency at long times ͑or low frequencies͒. On the other hand, the isotropic nature of the n-renormalized Rouse model suggests that the intermolecular contribution tends to prevail on long-time scales ͑or low frequencies͒. Actually, theoretical estimations and the analysis of experimental spin-lattice relaxation data indicate that the intermolecular contribution to proton NMR measurands is no longer negligible for times longer than 10 −7 s-10 −6 s corresponding to frequencies below the megahertz regime. Interpretations not taking this fact into account need to be reconsidered. The systematic investigation of intermolecular interactions in long-time/low frequency proton NMR promises the revelation of the dynamic features of segment displacements relative to each other in polymer melts.
Matrix effects on the diffusion of long polymer chains
Macromolecules, 1986
Macromolecules purification. All NMR spectra were obtained on a Varian XL-400 spectrometer operating a t 399.93 MHz for 'H and 100.56 MHz for 13C a t a probe temperature of 18 & 1 "C. Unless otherwise indicated, spectra were obtained for 0.4 M solutions of hydrocarbon in the appropriate solvent. Heteronuclear shift-correlated spectra were obtained with a version of this experiment that provides 'H-'H decoupling.21 Typical spectra were obtained with a 240-Hz spectral width in the fl ('H) domain and 2000-Hz spectral width for fz (13C). Forty time increments were used with zero-filling in 256 in fi while 2048 data points were collected in fz with zero-filling to 4096. Sixty-four transients were collected for each time increment and, with a relaxation delay of 1.0 s between increments, total measuring time was 1.1 h. Pseudoecho processingz2 was applied in both domains to ensure maximum resolution. Repeat measurements indicated that 'H chemical shifts could usually be determined with a precision of 0.005 ppm, Le., 2 Hz. Acknowledgment. Financial support from the Natural Sciences and Engineering Research Council of Canada in the form of a strategic grant (W.
Dynamics of a polymer chain confined in a membrane
The European Physical Journal E, 2011
We present a Brownian dynamics theory with full hydrodynamics (Stokesian dynamics) for a Gaussian polymer chain embedded in a liquid membrane which is surrounded by bulk solvent and walls. The mobility tensors are derived in Fourier space for the two geometries, namely, a free membrane embedded in a bulk fluid, and a membrane sandwiched by the two walls. Within the preaveraging approximation, a new expression for the diffusion coefficient of the polymer is obtained for the free membrane geometry. We also carry out a Rouse normal mode analysis to obtain the relaxation time and the dynamical structure factor. For large polymer size, both quantities show Zimm-like behavior in the free membrane case, whereas they are Rouse-like for the sandwiched membrane geometry. We use the scaling argument to discuss the effect of excluded volume interactions on the polymer relaxation time.